On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

TL;DR

This paper explores the algebraic structure of weighted zero-sum sequences over finite abelian groups, constructing transfer homomorphisms from norm monoids in Galois number fields and binary quadratic forms to analyze their properties.

Contribution

It introduces transfer homomorphisms from norm monoids and quadratic form monoids to weighted zero-sum sequence monoids, advancing the understanding of their algebraic and arithmetic properties.

Findings

Transfer homomorphisms connect norm monoids to weighted zero-sum sequences.

Analysis of algebraic properties of weighted zero-sum sequence monoids.

Insights into arithmetic behavior of these monoids in number theory contexts.

Abstract

Let $G$ be an additive finite abelian group and $\Gamma \subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($\Gamma$-)weighted zero-sum sequence if there are $\gamma_1, \ldots, \gamma_{\ell} \in \Gamma$ such that $\gamma_1 (g_1) + \ldots + \gamma_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $\Gamma$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.